Whitehead product

Definition

Given elements f ∈ π k ( X ) , g ∈ π l ( X ) , the Whitehead bracket

is defined as follows:

The product S k × S l can be obtained by attaching a ( k + l ) -cell to the wedge sum

the attaching map is a map

Represent f and g by maps

and

then compose their wedge with the attaching map, as

The homotopy class of the resulting map does not depend on the choices of representatives, and thus one obtains a well-defined element of

Grading

Note that there is a shift of 1 in the grading (compared to the indexing of homotopy groups), so π k ( X ) has degree ( k − 1 ) ; equivalently, L k = π k + 1 ( X ) (setting L to be the graded quasi-Lie algebra). Thus L 0 = π 1 ( X ) acts on each graded component.

Properties

The Whitehead product is bilinear, graded-symmetric, and satisfies the graded Jacobi identity, and is thus a graded quasi-Lie algebra; this is proven in Uehara & Massey (1957) via the Massey triple product.

If f ∈ π 1 ( X ) , then the Whitehead bracket is related to the usual conjugation action of π 1 on π k by

where g f denotes the conjugation of g by f . For k = 1 , this reduces to

which is the usual commutator.

The relevant MSC code is: 55Q15, Whitehead products and generalizations.